Radiation conditions for the difference schrödinger operators

Shaban, W.; Vainberg, Boris

doi:10.1080/00036810108841007

Cited by 65 publications

(90 citation statements)

References 5 publications

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“…• The approach used in this work has been previously used by the authors in different circumstances in [21,22] (see also [28]). Its idea originates from the paper [30] of the second author.…”

Section: Remarks and Acknowledgmentsmentioning

confidence: 99%

On the Structure of Eigenfunctions Corresponding to Embedded Eigenvalues of Locally Perturbed Periodic Graph Operators

Kuchment

Vainberg

2006

Commun. Math. Phys.

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The article is devoted to the following question. Consider a periodic self-adjoint difference (differential) operator on a graph (quantum graph) G with a co-compact free action of the integer lattice Z n . It is known that a local perturbation of the operator might embed an eigenvalue into the continuous spectrum (a feature uncommon for periodic elliptic operators of second order). In all known constructions of such examples, the corresponding eigenfunction is compactly supported. One wonders whether this must always be the case. The paper answers this question affirmatively. What is more surprising, one can estimate that the eigenmode must be localized not far away from the perturbation (in a neighborhood of the perturbation's support, the width of the neighborhood determined by the unperturbed operator only).The validity of this result requires the condition of irreducibility of the Fermi (Floquet) surface of the periodic operator, which is expected to be satisfied for instance for periodic Schrödinger operators.

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Section: Remarks and Acknowledgmentsmentioning

confidence: 99%

On the Structure of Eigenfunctions Corresponding to Embedded Eigenvalues of Locally Perturbed Periodic Graph Operators

Kuchment

Vainberg

2006

Commun. Math. Phys.

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“…The special feature of this paper is related to the radiation condition for the discrete problem which was found quite recently [1], [2] and is not widely known. We do not discretise the continuous radiation condition (4).…”

Section: Introductionmentioning

confidence: 99%

“…Equation (11) implies the following relation between ℓ and k The radiation condition for the difference equation (6) can be found in [1] and [2]. The form of the radiation condition depends essentially on the value of k 2 in the continuous spectrum Sp(−∆) = [0, 4n].…”

Section: Introductionmentioning

confidence: 99%

“…One can reduce the problem to the whole space R n by extending function u evenly through the boundary x n = 0 for x n < 0. Then equations (1)- (2) are replaced by the Schrödinger equation…”

Section: Introductionmentioning

confidence: 99%

“…We do not discretise the continuous radiation condition (4). Instead, we use the exact radiation condition for the discrete Schrödinger equation (6) that provides a unique solvability of the discrete problem on the lattice [1], [2]. Thus, the problem from the very beginning is discrete and, as we will show, is well-posed for a fixed step h. This makes the method differ from other approaches such as the linear sampling method [3]- [6], the least square method [7]- [8], or the factorization method [9] (see [10]- [11] for review of other methods).…”

Section: Introductionmentioning

confidence: 99%

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The inverse scattering problem for the difference operators: The decomposition method

Godin

Vainberg

2008

J. Phys.: Conf. Ser.

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Abstract. The inverse scattering problem of determining the boundary impedance from knowledge of the time harmonic incident wave and the far-field pattern of the scattered wave is considered. We solve the finite difference Helmholtz equation subject to the exact radiation condition for the discrete problem. The approach is decomposed into two steps. First, we reduce the problem to a well-posed system of linear equations for a modified potential. We next find the boundary impedance using the modified potential through an explicit formula. Thus, the computational part of the nonlinear problem of reconstruction of the boundary impedance is reduced to the solution of a linear system. Numerical examples are given to demonstrate efficiency of the new method.

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Fermi isospectrality for discrete periodic Schrödinger operators

Liu

2023

Comm Pure Appl Math

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Let normalΓ=q1double-struckZ⊕q2double-struckZ⊕…⊕qddouble-struckZ$\Gamma =q_1\mathbb {Z}\oplus q_2 \mathbb {Z}\oplus \ldots \oplus q_d\mathbb {Z}$, where ql∈Z+$q_l\in \mathbb {Z}_+$, l=1,2,…,d$l=1,2,\ldots ,d$, are pairwise coprime. Let normalΔ+V$\Delta +V$ be the discrete Schrödinger operator, where Δ is the discrete Laplacian on double-struckZd$\mathbb {Z}^d$ and the potential V:Zd→double-struckC$V:\mathbb {Z}^d\rightarrow \mathbb {C}$ is Γ‐periodic. We prove three rigidity theorems for discrete periodic Schrödinger operators in any dimension d≥3$d\ge 3$: If at some energy level, Fermi varieties of two real‐valued Γ‐periodic potentials V and Y are the same (this feature is referred to as Fermi isospectrality of V and Y), and Y is a separable function, then V is separable; If two complex‐valued Γ‐periodic potentials V and Y are Fermi isospectral and both V=⨁j=1rVj$V=\bigoplus _{j=1}^rV_j$ and Y=⨁j=1rYj$Y=\bigoplus _{j=1}^r Y_j$ are separable functions, then, up to a constant, lower dimensional decompositions Vj$V_j$ and Yj$Y_j$ are Floquet isospectral, j=1,2,…,r$j=1,2,\ldots ,r$; If a real‐valued Γ‐potential V and the zero potential are Fermi isospectral, then V is zero. In particular, all conclusions in (1), (2) and (3) hold if we replace the assumption “Fermi isospectrality” with a stronger assumption “Floquet isospectrality”.

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Radiation conditions for the difference schrödinger operators

Cited by 65 publications

References 5 publications

On the Structure of Eigenfunctions Corresponding to Embedded Eigenvalues of Locally Perturbed Periodic Graph Operators

On the Structure of Eigenfunctions Corresponding to Embedded Eigenvalues of Locally Perturbed Periodic Graph Operators

The inverse scattering problem for the difference operators: The decomposition method

Fermi isospectrality for discrete periodic Schrödinger operators

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